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56 111 METRIC SPACES
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xeV, d'(f(a\f(x)}<*l2 and d'(s(^\9(x))<^ Then for *eV,
f(x)^g(x), otherwise we would have d'(f(a\g(a))<& by the triangle inequality. |
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(3.15.2) ("Principle of extension of identities") Let f, g be two contin-
uous mappings of a metric space E into a metric space E'; iff(x)-g(x) for all points xofa dense subset A in E, thenf-g.
For the set of points x where f(x)=g(x) is closed by (3.15.1) and
contains A. |
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(3.15.3) Let f9 g be two continuous mappings of a metric space E into the
extended real line E. The set P of the points xeE such that f(x) ^g(x) is closed in E.
We prove again E - P is open. Suppose/(a) > g(a), and let P e E be such
that/(a) > jff > g(a) (cf. (2.2.16) and the definition of I in Section 3.3). The inverse image V by/of the open interval ]/?, + oo] is a neighborhood of a by (3.11.1); so is the inverse image W by g of the open interval [-00, /?[. Hence V n W is a neighborhood of a by (3.6.3), and for x e V n W,/(;c) > )8 > g(x). Q.E.D. |
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(3.15.4) (" Principle of extension of inequalities ") Let f, g be two contin-
uous mappings of a metric space E into the extended real line E; iff(x) ^ g(x) for all points x of a dense subset A ofE, thenf(x) < g(x)for all x e E.
The proof follows from (3.15.3) as (3.15.2) from (3.15.1).
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(3.15.5) Let A be a dense subset of a metric space E, and fa mapping of A
into a metric space E'. In order that there exist a continuous mapping f of E into E', coinciding with fin A, a necessary and sufficient condition is that, for any x e E, the limit lim f(y) exist in E1; the continuous mapping J is then unique. )>-+*, ye A
As any x e E belongs to A, we must have/(#) = lim J(y) by (3.13,5),
hence f(x) = lim f(y)\ this shows the necessity of the condition and the
y-+x,yeA
fact that if the continuous mapping / exists, it is unique (this follows also
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